Derivative of a constant function via the limit definition

Since $f(x) = 7$ for all $x$, it follows that $f(x+h) = 7$ as well, because the function outputs $7$ regardless of its input. Substituting into the difference quotient gives $\dfrac{f(x+h)-f(x)}{h} = \dfrac{7-7}{h} = \dfrac{0}{h}$.

For any $h \neq 0$, $\dfrac{0}{h} = 0$, so the difference quotient is identically $0$ before taking the limit. Therefore $f'(x) = \lim_{h\to 0} 0 = 0$, confirming that the derivative of a constant function is always $0$.

The numerator is always zero. Since f(x) equals seven for every input, f(x plus h) also equals seven, so subtracting gives seven minus seven, which is zero. This zero numerator, divided by h, makes the whole difference quotient zero before any limit is taken.